This directly refers to the Futures End Booster Gold issue so SPOILERS AHEAD for those who haven't read it. Oh and BG vol. 2.
Okay here we go.
The issue ends with the younger Booster caving to his tormentors' demands to save an alternate Michelle's life. He agrees to tell the mystery foe(s?) whatever they want and take them to Vanishing Point. Is this out of character for Booster Gold? Well he's always had a problem letting go going back to turning to gambling to save his mother even though it risked any hope they had at a better life.
Granted the new 52 version hasn't been fleshed out too much but he seems the same as the old one. Which makes sense as he's one of the few characters we saw move into the post-Flashpoint world. One of the themes explored in volume 2 of his series was Booster trying to come to grips with the hard choices a Time Master has to face. This usually centered around letting people die because they were supposed to like Ted Kord. Booster himself didn't solve that problem as Ted accepted his fate and took the choice out of his hands. When Coast City was in danger he wavered again trying to save lives but ultimately being unable to do so. His future self actually had to give him a literal push to get him out of harms' way and left Michelle's' boyfriend Drew to die like he was supposed to.
This struggle to accept the rules of being a Time Master often brought him into conflict with Rip. Now Rip doesn't like it but accepts that some things can't be changed and there are sacrifices that must be made. He actually goes steps further by being willing to torture to get information and it's been implied that he will kill if need be. This isn't the normal heroic thought which naturally brought him into conflict with Superman and Hal during their search for Batman. During that time the two opposing ideals obviously weighed heavy on Booster. Should he agree with the heroes he always wanted to be like or do what he knows from experience is right despite knowing lives will be lost?
Michelle has always been someone special for Booster, his only link to his time and his twin. Jurgens never wrote the relationship as stereotypical as other comic writers make twins but that link is still a important one. This version of Booster already lost his sister and by saying no would be directly responsible for another versions death which isn't an easy choice anyway you slice it.
We know that older Booster will accept these rules but he's not there yet. I think what's so upsetting about him giving in is that it feels like a betrayal. To Rip, to his job as a Time Master and as a hero since he knows no good can come from this. Since we know whatever storyline is to come will be based of his actions it doesn't paint him in the best of lights. The choice might haunt the character for years to come as more than one earth seems threatened because of this. But here's something else to take in mind about Booster...he's kind of a con man. I know how that sounds but for lack of a better term it's what I choose. This is a guy that in one timeline came up with schemes along side Ted. Someone Winick and Jurgens were shaping into a leader. That threw his image away to stop the Manhunters, defeat Mr. Mind and when he became a Time Master. Booster can be tricky. He was the greatest hero you never heard of for a reason. Everyone underestimates him.